Desbalanceamento

Desbalanceamento#

Nesta seção o equacionamento realizado considerou a aplicação de uma massa de desbalanço excitando o sitema conforme proposto no livro texto da disciplina, “Rotordynamics Prediction in Engineering, Second Edition”, Lalanne and Ferraris [1998].

Foram calculadas as amplitudes de vibração em função da velocidade de rotação

Importando módulos e configurando o ambiente#

'''
    Módulos standard
'''
from collections import namedtuple

'''
    Bibliotecas de terceira parte
'''
import numpy as np

'''
    Biblioteca criada para a realização das análises dinâmicas
    no contexto da disciplina
'''
import rotor_analysis as rd

'''
    Configurando pint.Quantity
'''
Q_ = rd.Q_

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
Cell In[1], line 15
      9 import numpy as np
     11 '''
     12     Biblioteca criada para a realização das análises dinâmicas
     13     no contexto da disciplina
     14 '''
---> 15 import rotor_analysis as rd
     17 '''
     18     Configurando pint.Quantity
     19 '''
     20 Q_ = rd.Q_

File /opt/hostedtoolcache/Python/3.12.5/x64/lib/python3.12/site-packages/rotor_analysis/__init__.py:54
     52 from .rotordynamics import Disc, Shaft, Rotor
     53 from .utilities import Material, Q_
---> 54 from .results import (
     55     add_secondary_yaxis,
     56     campbell_diagram_axial_forces,
     57     create_video_from_frames,
     58     FFT,
     59     interactive_orbit,
     60     interactive_orbit_campbell,
     61     interactive_orbit_campbell_async,
     62     interactive_orbit_fixed_speed,
     63     plot_vibration_amplitude,
     64     save_orbit_frames,
     65     update_circle_and_sine,
     66 )
     69 __all__ = [
     70     "add_secondary_yaxis",
     71     "campbell_diagram_axial_forces",
   (...)
     85     "update_circle_and_sine",
     86 ]

File /opt/hostedtoolcache/Python/3.12.5/x64/lib/python3.12/site-packages/rotor_analysis/results.py:27
     25 import plotly.graph_objects as go
     26 import ipywidgets as widgets
---> 27 import cv2
     29 from typing import Any, Callable, Dict, Tuple
     30 from plotly.subplots import make_subplots

ModuleNotFoundError: No module named 'cv2'

Definindo parâmetros#

# Material instances and properties
steel = rd.Material(name='Steel',
                    density=Q_(7800, 'kg/m^3'),
                    young_modulus=Q_(2e11,"Pa"))

# Shaft
L = Q_(0.4, 'm')
shaft = rd.Shaft(outer_radius=Q_(0.01, 'm'),
                 inner_radius=Q_(0.0, 'm'),
                 length=L,
                 material=steel)

# Disc
disc = rd.Disc(outer_radius=Q_(0.150, 'm'),
               inner_radius=Q_(0.010, 'm'),
               length=Q_(0.030, 'm'),
               material=steel,
               coordinate=L/3)

# Rotor
rotor = rd.Rotor(shaft, disc)
speed_range = np.linspace(0, 9000, 101)

# Unbalance
Unbalance = namedtuple(
    'Unbalance',
    ['mass', 'radius']
)
'''
 Numerical data Reference:
     Rotordynamics Prediction in Engineering, Second Edition, by Michel
     Lalanne and Guy Ferraris, Chapter 2: Monorotors, Subsection 2.1.6, page 17.
'''
unbal = Unbalance(
    mass=Q_(1e-4, 'kg'),
    radius=Q_(0.15, 'm')
)

def A1_unbal(speed, *args):
    '''Function to compute the displacemente aplitude A1 of the unbalanced rotor at
    some speed. 
    
    Args:
        speed (float): Rotational speed in RPM.
        args: unbalance named tuple with mass and radius as properties

    Reference:
        Rotordynamics Prediction in Engineering, Second Edition, by Michel
        Lalanne and Guy Ferraris equation 
    '''
    unbalance = args[0]
    k1, k2 = rotor.stiffness
    m = rotor.mass
    spin = speed / 60 * 2 * np.pi
    a = rotor.a
    div = (k1 - m * spin**2) * (k2 - m * spin**2) - a**2 * spin**4
    return (k2 - (m + a) * spin**2) * unbalance.mass.m * unbalance.radius.m * (spin)**2 * rotor.f(rotor.discs[0].coordinate.m) / div

# Computing the values
values = []
for speed in speed_range:
    values.append(abs(A1_unbal(speed, unbal)))

# Generating the Standard Campbell Diagram
campbell_fig = rotor.plot_Campbell()

# Call the function to add the secondary y-axis
rd.add_secondary_yaxis(campbell_fig, values)

# Show the updatedvfigure
campbell_fig.show()

A seguir é calculado o valor da amplitude de vibração quando a rotação tende ao infinto.

'''
 Reference:
     Rotordynamics Prediction in Engineering, Second Edition, by Michel
     Lalanne and Guy Ferraris equation 2.134
'''
unbal.mass.m * rotor.discs[0].outer_radius.m * rotor.f(rotor.discs[0].coordinate.m) \
/ (rotor.a - rotor.mass)

np.float64(-1.1371292754494908e-06)

Órbita#

Após o cômputo da amplitude da resposta ao desbalanço, plota-se os deslocamentos \(q1\) e \(q2\) formando-se assim a orbita descrita pelo centro do eixo. No caso abaixo temos a órbita simulando o encoder posicionado em q2 (no topo do eixo), e o desbalanço posicionado na chaveta.

Combinando os gráficos anteriormente gerados, temos:

orbit_plot = rd.interactive_orbit_campbell(campbell_fig,
                                           A1_unbal,
                                           A1_unbal,
                                           unbal,
                                           initial_speed=2700,
                                           max_amplitude=2e-05)
orbit_plot.show()
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